We have the following recurrence formulae [1]}:
\(K_{\nu +1}(t)=\frac{2\nu }{t}K_{\nu }(t)+K_{\nu -1}(t)\)  and \(K_{\nu -1}(t)+K_{\nu +1}(t)=-2\partial _t K_{\nu }(t)\) .
It follows that \(K_{\nu }(t)\)  is monotonically decreasing and that \(\partial _t K_{\nu }(t)=-K_{\nu -1}(t)-\frac{\nu }{t}K_{\nu }(t)\) .
Further, we have
\(\partial _t \log K_{\nu }(t) &= - \frac{K_{\nu -1}(t)}{K_{\nu }(t)} -\frac{\nu }{t} =: -R_\nu (t) -\frac{\nu }{t},\\\partial ^2_t \log K_{\nu }(t) &= - \frac{1}{K^2_{\nu }(t)}\left(K^2_{\nu -1}(t)-K_{\nu -2}(t)K_{\nu }(t)\right) -\frac{1}{t}\frac{K_{\nu -1}(t)}{K_{\nu }(t)}+\frac{\nu }{t^2}\nonumber \\&=: -S_\nu (t) - \frac{1}{t}R_\nu (t) + \frac{\nu }{t^2}.\) 
